Churchlands, Representational Disconcert via State-Space Physics & Phase-Space Mathematics

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One thought on “Churchlands, Representational Disconcert via State-Space Physics & Phase-Space Mathematics

  1. Let a dynamic system be specified by a function f(t, *). If ‘a’ is a point in the phase space (a space in which all possible states of a system is represented, with each possible state of the system corresponding to one unique point in the phase space), so that a state of the system at a certain time, then f(0, a) = a and for a positive value of t, f(t, a) is the result of the evolution of this state after t units of time. An attractor, is then a subset A of the phase space characterized by three conditions:

    a) A is forward invariant under f: if a is an element of A then so is f(t, a), ∀ t > 0.

    b) There exists a neighborhood of A, called the basin of attraction for A and denoted B(A), which consists of all points b that “enter A in the limit t → ∞”. More formally, B(A) is the set of all points b in the phase space with the following property:

    For any open neighborhood N of A, there is a large positive constant T such that f(t,b) ∈ N ∀ ℜ t > T.

    c) There is no proper subset of A having the first two properties.

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